Colloque - Kathrin Smetana : Certified Randomized Model Order Reduction Methods for High-Dimensional Approximation

Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Kathrin Smetana : Certified Randomized Model Order Reduction Methods for High-Dimensional ApproximationKathrin SmetanaTenure-track Assistant professor in the Department of Mathematical Sciences at the Stevens Institute of Technology, Hoboken, New Jersey, USARésuméIn this talk, we present randomized methods that provide high-probability guarantees for the accuracy of reduced order approximations of parametric partial differential equations (PDEs) with high-dimensional parameter sets. The underlying philosophy is to combine classical reduced basis and greedy approximation ideas with concentration phenomena and data-dependent sampling to obtain certified approximations in high dimensions.We first present non-asymptotic error bounds for the Proper Orthogonal Decomposition (POD) under the sole assumption that the parameter-to-solution map is uniformly bounded for almost all parameter values. In contrast to existing results, the leading term in our bounds is governed by the sum of the neglected eigenvalues and scales inversely with the number of samples, thereby allowing one to exploit rapid eigenvalue decay. The resulting estimates are independent of the dimension of the parameter space. Consequently, even a modest number of samples can be sufficient for the empirical POD approximation to perform comparably to the ideal POD constructed from the full parameter distribution, including in infinite-dimensional parameter settings.